The Inverse of Trigonometric Functions

1. The Inverse of Trigonometric Functions Find the Exact Value of all Trigonometric Functions Find the Approximate Value of the Inverse of Trigonometric Functions

2. Review Chapter 4 We discussed inverse functions and we noted that if a function is one to one it will have an inverse function We also discussed that if a function is not one to one it may be possible to restrict the domain in some manner so that the restricted function is one to one

3. Properties of a one to one function 1. f-1(f(x))=x for every x in the domain of f and f(f-1(x)) =x for every x in the domain of f-1 • Domain of f = range of f-1, and the range of f = domain of f-1 • The graph of f and graph of f-1 are symmetric with respect to the line y=x • If a function y=f(x) has an inverse function, the equation of the inverse function is x= f(y). The solution of this equation is y=f-1(x)

4. The inverse sine function Graph of the function Because every horizontal line y=b where b is between -1 and intersects the graph of y=sin x Infinitely many times it follows from the horizontal line test that the function is not one to one. However if we restrict the domain of the function between the restricted function is one to one and hence will have an inverse function

5. How do we find the inverse An equation fro the inverse of y=f(x) = sin x is obtained by changing x and y. The explicit form is called the inverse sine of x and is denoted by y = f-1(x) and sin-1 x y= sin-1 x means x = sin y Where -1≤ x ≤ 1 and

6. Finding the exact Value of an Inverse sine function Find the exact value of: sin-1 1 Find the exact value of: sin-1

7. Finding the Approximate Value of an Inverse Sine Find the approximate value of sin-1 Round your answer to the nearest hundredth of a radian Find the approximate value of sin-1 Round your answer to the nearest hundredth of a radian

8. Showing they are inverses f-1(f(x))= sin-1(sin x) = x, where f(f-1(x)) = sin (sin-1 x) = x where -1≤ x≤ 1

9. Finding the exact value of a compositefunction of sine

10. Inverse Cosine Graph cos x We are going to have to restrict the domain of the cosine also but differently from the sine function. Why?

11. Inverse cosine We need to interchange our x and y to find the inverse of the cosine y = cos-1 x means x= cos y where -1≤x≤1 and 0≤ y ≤ π

12. Finding the exact value Find the exact value of: cos-1 0 Find the exact value of: cos-1

13. Definition of inverse f-1(f(x)) = cos-1(cos x) = x where 0≤x≤π f(f-1(x)) = cos(cos-1 x) = x where -1≤x ≤ 1

14. Finding the exact value of a composite function Find the exact value of :

15. The inverse tangent function Again for the tangent function we must restrict the domain of y=tan x to the interval of y = tan-1x means x = tan y Where -∞<x<∞ and

16. Finding the exact value of an inverse tangent function Find the exact value of: tan-1 = 1 Find the exact value of: tan-1( )

17. The remaining Inverse trigonometric functions y = sec-1x means x= sec y where |x|≥ 1 and 0≤ y ≤ π, y≠ y = csc-1x means x=csc y where |x|≥ 1 and y= cot-1 x means x= cot y where -∞<x<∞ and 0<y<π

18. Classwork Page 423 examples 13-44 every fourth problem Page 429 examples 9-41 first column

19. Homework Page 429 10-54 second column